Presentation slides

Stochastic Polynomial approximation of PDEs
with random coefficients
Fabio Nobile
CSQI-MATHICSE, EPFL, Switzerland
and MOX, Politecnico di Milano, Italy
Joint work with: R. Tempone, E. von Schwerin (KAUST)
L. Tamellini, G. Migliorati (MOX, Politecnico Milano), J. Beck (UCL)
WS: “Numer. Anal. of Multiscale Problems & Stochastic Modelling”
RICAM, Linz, December 12-16, 2011
Italian project FIRB-IDEAS (’09) Advanced Numerical Techniques for Uncertainty Quantification in Engineering and Life
Science Problems
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
1
Outline
1
Elliptic PDE with random coefficients
2
Stochastic multivariate polynomial approximation
Galerkin projection
Collocation on sparse grids
Optimized algorithms
Numerical results
3
Polynomial approximation by discrete projection on random points
4
Conclusions
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
3
Elliptic PDE with random coefficients
Outline
1
Elliptic PDE with random coefficients
2
Stochastic multivariate polynomial approximation
Galerkin projection
Collocation on sparse grids
Optimized algorithms
Numerical results
3
Polynomial approximation by discrete projection on random points
4
Conclusions
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
4
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
Let (Ω, F, P) be a complete probability space. Consider
(
− div(a(ω, x)∇u(ω, x)) = f (x) x ∈ D, ω ∈ Ω,
L(u) = F ⇔
u(ω, x) = 0
x ∈ ∂D, ω ∈ Ω
where
a : Ω × D → R is a second order random field such that
a ≥ amin > 0 a.e. in Ω × D
f ∈ L2 (D) deterministic (could be stochastic as well,
f ∈ L2P (Ω) ⊗ L2 (D))
By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution
CP
u(ω, ·) ∈ V ≡ H01 (D), and kukV ⊗L2P ≤
kf kL2 (D)
amin
The uniform coercivity assumption can be relaxed to
E[amin (ω)−p ] < ∞ for all p > 0. Useful for lognormal random fields
(see e.g. [Babuska-N.-Tempone ’07, Garvis-Sarkis ’09, Gittelson ’10, Charrier ’11]).
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
5
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
Let (Ω, F, P) be a complete probability space. Consider
(
− div(a(ω, x)∇u(ω, x)) = f (x) x ∈ D, ω ∈ Ω,
L(u) = F ⇔
u(ω, x) = 0
x ∈ ∂D, ω ∈ Ω
where
a : Ω × D → R is a second order random field such that
a ≥ amin > 0 a.e. in Ω × D
f ∈ L2 (D) deterministic (could be stochastic as well,
f ∈ L2P (Ω) ⊗ L2 (D))
By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution
CP
u(ω, ·) ∈ V ≡ H01 (D), and kukV ⊗L2P ≤
kf kL2 (D)
amin
The uniform coercivity assumption can be relaxed to
E[amin (ω)−p ] < ∞ for all p > 0. Useful for lognormal random fields
(see e.g. [Babuska-N.-Tempone ’07, Garvis-Sarkis ’09, Gittelson ’10, Charrier ’11]).
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
5
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
Let (Ω, F, P) be a complete probability space. Consider
(
− div(a(ω, x)∇u(ω, x)) = f (x) x ∈ D, ω ∈ Ω,
L(u) = F ⇔
u(ω, x) = 0
x ∈ ∂D, ω ∈ Ω
where
a : Ω × D → R is a second order random field such that
a ≥ amin > 0 a.e. in Ω × D
f ∈ L2 (D) deterministic (could be stochastic as well,
f ∈ L2P (Ω) ⊗ L2 (D))
By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution
CP
u(ω, ·) ∈ V ≡ H01 (D), and kukV ⊗L2P ≤
kf kL2 (D)
amin
The uniform coercivity assumption can be relaxed to
E[amin (ω)−p ] < ∞ for all p > 0. Useful for lognormal random fields
(see e.g. [Babuska-N.-Tempone ’07, Garvis-Sarkis ’09, Gittelson ’10, Charrier ’11]).
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
5
Elliptic PDE with random coefficients
Assumption – random field parametrization
The stochastic coefficient a(ω, x) can be parametrized by a
finite number of independent random variables
a(ω, x) = a(y1 (ω), . . . , yN (ω), x)
We assume
Q that y has a joint probability density function
ρ(y) = Nn=1 ρn (yn ) : ΓN → R+
Then u(ω, x) = u(y1 (ω), . . . , yN (ω), x) is a deterministic function of
the random vector y.
Extensions to the case of infinitely many (countable) random
variables is possible provided the solution u(y1 , . . . , yn , . . . , x)
has suitable decay properties w.r.t. n.
(see e.g. [Cohen-Devore-Schwab, ’09,’10])
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
6
Elliptic PDE with random coefficients
Assumption – random field parametrization
The stochastic coefficient a(ω, x) can be parametrized by a
finite number of independent random variables
a(ω, x) = a(y1 (ω), . . . , yN (ω), x)
We assume
Q that y has a joint probability density function
ρ(y) = Nn=1 ρn (yn ) : ΓN → R+
Then u(ω, x) = u(y1 (ω), . . . , yN (ω), x) is a deterministic function of
the random vector y.
Extensions to the case of infinitely many (countable) random
variables is possible provided the solution u(y1 , . . . , yn , . . . , x)
has suitable decay properties w.r.t. n.
(see e.g. [Cohen-Devore-Schwab, ’09,’10])
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
6
Elliptic PDE with random coefficients
Assumption – random field parametrization
The stochastic coefficient a(ω, x) can be parametrized by a
finite number of independent random variables
a(ω, x) = a(y1 (ω), . . . , yN (ω), x)
We assume
Q that y has a joint probability density function
ρ(y) = Nn=1 ρn (yn ) : ΓN → R+
Then u(ω, x) = u(y1 (ω), . . . , yN (ω), x) is a deterministic function of
the random vector y.
Extensions to the case of infinitely many (countable) random
variables is possible provided the solution u(y1 , . . . , yn , . . . , x)
has suitable decay properties w.r.t. n.
(see e.g. [Cohen-Devore-Schwab, ’09,’10])
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
6
Elliptic PDE with random coefficients
Examples
Material with inclusions of random conductivity
a(ω, x) = a0 +
PN
n=N
yn (ω)1Ωn (x)
with yn ∼ uniform, lognormal, ...
Random, spatially correlated, material properties
a(ω, x) is ∞-dimensional random field
(e.g. lognormal), suitably truncated
a(ω, x) ≈ amin + e
PN
n=1 yn (ω)bn (x)
with yn ∼ N(0, 1), i.i.d.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
7
Elliptic PDE with random coefficients
Examples
Material with inclusions of random conductivity
a(ω, x) = a0 +
PN
n=N
yn (ω)1Ωn (x)
with yn ∼ uniform, lognormal, ...
Random, spatially correlated, material properties
a(ω, x) is ∞-dimensional random field
(e.g. lognormal), suitably truncated
random field with Lc=1/4
1
0.9
2.5
0.8
2
0.7
1.5
0.6
a(ω, x) ≈ amin + e
PN
n=1 yn (ω)bn (x)
0.5
1
0.4
0.5
0.3
0
0.2
−0.5
0.1
0
with yn ∼ N(0, 1), i.i.d.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
0
0.2
0.4
0.6
0.8
1
−1
RICAM, Linz, 12-16/12/2011
7
Elliptic PDE with random coefficients
Analytic regularity
Theorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10]
Assume y bounded (e.g. y ∈ ΓN ≡ [−1, 1]N )
Let iQ
= (i1 , . . . , iN ) ∈ NN and r = (r1 , . . . , rN ) > 0. Set
i
r = n rnin .
i
assume k 1a ∂∂yai kL∞ (D) ≤ ri uniformly in y
Then
i
k ∂∂yui kV ≤ C |i|!( log1 2 r)i
uniformly in y
u : ΓN → V is analytic and can be extended analyticially to
(
)
N
X
Σ = z ∈ CN :
rn |zn − ỹn | < log 2 for some ỹ ∈ ΓN
n=1
Better estimates on analyticity region can be obtained by complex analysis.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
8
Elliptic PDE with random coefficients
Analytic regularity
Theorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10]
Assume y bounded (e.g. y ∈ ΓN ≡ [−1, 1]N )
Let iQ
= (i1 , . . . , iN ) ∈ NN and r = (r1 , . . . , rN ) > 0. Set
i
r = n rnin .
i
assume k 1a ∂∂yai kL∞ (D) ≤ ri uniformly in y
Then
i
k ∂∂yui kV ≤ C |i|!( log1 2 r)i
uniformly in y
u : ΓN → V is analytic and can be extended analyticially to
(
)
N
X
Σ = z ∈ CN :
rn |zn − ỹn | < log 2 for some ỹ ∈ ΓN
n=1
Better estimates on analyticity region can be obtained by complex analysis.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
8
Stochastic multivariate polynomial approximation
Outline
1
Elliptic PDE with random coefficients
2
Stochastic multivariate polynomial approximation
Galerkin projection
Collocation on sparse grids
Optimized algorithms
Numerical results
3
Polynomial approximation by discrete projection on random points
4
Conclusions
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
10
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
Idea: approximate the response function u(y, ·) by global multivariate
polynomials. Since u(y, ·) is analytic, we expect fast convergence.
Let Λ ⊂ NN be an index set of cardinality |Λ| = M, and consider the
multivariate polynomial space
nQ
o
N
pn
PΛ (ΓN ) = span
y
,
with
p
=
(p
,
.
.
.
,
p
)
∈
Λ
1
N
n=1 n
Polynomial approximation
find M particular solutions up ∈ V , ∀p ∈ Λ and build
X
uΛ (y, x) =
up (x)y1p1 y2p2 · · · yNpN
p∈Λ
Compute statistics of functionals J(u), J ∈ H −1 (D) of the
solution as E[J(u)] ≈ E[J(uΛ )]
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
11
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
Idea: approximate the response function u(y, ·) by global multivariate
polynomials. Since u(y, ·) is analytic, we expect fast convergence.
Let Λ ⊂ NN be an index set of cardinality |Λ| = M, and consider the
multivariate polynomial space
nQ
o
N
pn
PΛ (ΓN ) = span
y
,
with
p
=
(p
,
.
.
.
,
p
)
∈
Λ
1
N
n=1 n
Polynomial approximation
find M particular solutions up ∈ V , ∀p ∈ Λ and build
X
uΛ (y, x) =
up (x)y1p1 y2p2 · · · yNpN
p∈Λ
Compute statistics of functionals J(u), J ∈ H −1 (D) of the
solution as E[J(u)] ≈ E[J(uΛ )]
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
11
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
Idea: approximate the response function u(y, ·) by global multivariate
polynomials. Since u(y, ·) is analytic, we expect fast convergence.
Let Λ ⊂ NN be an index set of cardinality |Λ| = M, and consider the
multivariate polynomial space
nQ
o
N
pn
PΛ (ΓN ) = span
y
,
with
p
=
(p
,
.
.
.
,
p
)
∈
Λ
1
N
n=1 n
Polynomial approximation
find M particular solutions up ∈ V , ∀p ∈ Λ and build
X
uΛ (y, x) =
up (x)y1p1 y2p2 · · · yNpN
p∈Λ
Compute statistics of functionals J(u), J ∈ H −1 (D) of the
solution as E[J(u)] ≈ E[J(uΛ )]
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
11
Stochastic multivariate polynomial approximation
Examples of pol. spaces:
N = 2,
pn ≤ w
Total Degree (TD)
16
16
14
14
12
12
10
10
p2
Tensor
product:
p2
Tensor Product (TP)
8
6
4
4
2
Total
P
n
8
6
0
Q
(pn + 1)
≤w +1
n
2
4
6
8
p1
10
12
14
0
16
16
14
14
12
12
10
10
p2
p2
pn ≤ w
0
2
4
6
8
p1
10
12
14
16
Smolyak (SM)
16
8
6
4
4
2
Smolyak:
P
n
8
6
0
degree:
2
0
Hyperbolic Cross (HC)
Hyperbolic
cross:
p = 16
f (p) =
2
0
2
Fabio Nobile (EPFL & PoliMi)
4
6
8
p1
10
12
14
16
0
0
2
4
6
8
p1
Polynomial approximation of SPDEs
10
12
14
f(pn ) ≤ f (w )

0,
1,

dlog (p)e,
2
p=0
p=1
p≥2
16
RICAM, Linz, 12-16/12/2011
12
Stochastic multivariate polynomial approximation
Galerkin projection
Galerkin projection
[Ghanem-Spanos, Karniadakis et al, Matthies-Keese, Schwab-Todor et al., Knio-Le Maı̂tre et
al,Babuska et al.,. . . ]
Project the equation L(y)(u) = F onto the subspace V ⊗ PΛ (Γ)
Let {ψj }M
j=1 be an orthonormal basis w.r.t. the probability density
P
ρ(y). Expand uΛ (y) on the basis: uΛ (y) = M
j=1 uj ψj (y)
Galerkin formulation
Find uj ∈ V , j = 1, . . . M s.t.
M
h
i
X
E L(y)(
uj ψj )ψi = E[Fψi ],
i = 1, . . . , M
j=1
This approach leads to solving M coupled deterministic
problems; difficult to assemble and need good preconditioners.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
13
Stochastic multivariate polynomial approximation
Collocation on sparse grids
Collocation approach
[Smolyak ’63, Griebel et al ’98-’03-’04, Barthelmann-Novak-Ritter ’00, Hesthaven-Xiu ’05,
N.-Tempone-Webster ’08, Zabaras et al ’07]
1
2
3
e
Choose a set of points y(j) ∈ Γ, j = 1, . . . , M
Compute the solutions uj ∈ V : L(y(j) )(uj ) = F
Pe
Interpolate the obtained values: uΛ (y) = M
j=1 uj φj (y).
φj ∈ PΛ (Γ): suitable combinations of Lagrange polynomials
e uncoupled deterministic problems
Always leads to solving M
e of points needed is larger than the dimension M
The number M
of the polynomial space (Except for tensor product spaces).
Tensor grids are impractical in high dimension (curse of
dimensionality)
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
15
Stochastic multivariate polynomial approximation
Collocation on sparse grids
Collocation approach
[Smolyak ’63, Griebel et al ’98-’03-’04, Barthelmann-Novak-Ritter ’00, Hesthaven-Xiu ’05,
N.-Tempone-Webster ’08, Zabaras et al ’07]
1
2
3
e
Choose a set of points y(j) ∈ Γ, j = 1, . . . , M
Compute the solutions uj ∈ V : L(y(j) )(uj ) = F
Pe
Interpolate the obtained values: uΛ (y) = M
j=1 uj φj (y).
φj ∈ PΛ (Γ): suitable combinations of Lagrange polynomials
e uncoupled deterministic problems
Always leads to solving M
e of points needed is larger than the dimension M
The number M
of the polynomial space (Except for tensor product spaces).
Tensor grids are impractical in high dimension (curse of
dimensionality)
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
15
Stochastic multivariate polynomial approximation
Collocation on sparse grids
Collocation on a (generalized) Sparse Grid
Let i = [i1 , . . . , iN ] ∈ NN+ and m(i) : N+ → N+ an increasing function
1
m(in )
1D polynomial interpolant operators: Un
We use either
on m(in ) abscissas.
Clenshaw-Curtis (extrema on Chebyshev polynomials)
Gauss points w.r.t. the weight
Q ρn , assuming that the probability
density factorizes as ρ(y) = N
n=1 ρn (yn )
2
3
m(i )
m(i )
m(i −1)
m(0)
Detail operator: ∆n n = Un n − Un n , Un
= 0.
N
Sparse grid approximation: on an index set Λ ⊂ N
uΛ =
N
XO
n)
∆m(i
[u]
n
i∈Λ n=1
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
16
Stochastic multivariate polynomial approximation
Collocation on sparse grids
By choosing properly the function m and the set Λ one can obtain a
polynomial approximation in any given multivariate polynomial
space ([Back-N.-Tamellini-Tempone, 2010])
Examples of sparse grids:
Fabio Nobile (EPFL & PoliMi)
N = 2,
max. polynomial degree p = 16
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
17
Stochastic multivariate polynomial approximation
Collocation on sparse grids
By choosing properly the function m and the set Λ one can obtain a
polynomial approximation in any given multivariate polynomial
space ([Back-N.-Tamellini-Tempone, 2010])
Examples of sparse grids:
N = 2,
max. polynomial degree p = 16
Tensor Product (TP)
Total Degree (TD)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
y2
y2
0.2
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
Hyperbolic Cross (HC)
−1
−1
1
−0.8
−0.6
−0.4
−0.2
0
0.2
y1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
y2
1
0
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
Fabio Nobile (EPFL & PoliMi)
0.4
0.6
0.8
1
−1
−1
0.8
1
0
−0.2
−0.8
0.6
Smolyak CC (SM)
1
−1
−1
0.4
Smolyak Gauss (SM)
1
y2
y2
0
−0.2
−0.8
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
Polynomial approximation of SPDEs
−1
−1
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
RICAM, Linz, 12-16/12/2011
17
Stochastic multivariate polynomial approximation
Collocation on sparse grids
A simple numerical test
(
−(a(x, ω)u(x, ω)0 )0 = sin(πx), x ∈ (0, 1),
u(0, ω) = u(1, ω) = 0
1D problem:
Diffusion coefficient a(x, ω) = e γ(x,ω) : lognormal with either
exponential or Gaussian covariance
γ(x, ω): stationary Gaussian random field, mean=0; std=1;
correlation length: 0.3
Karhunen-Loève expansion
3
2
2
1
1
log − conductivity
log − conductivity
3
0
−1
0
−1
−2
−2
−3
−3
0
0.1
0.2
0.3
0.4
0.5
x
0.6
Fabio Nobile (EPFL & PoliMi)
0.7
0.8
0.9
1
0
0.1
0.2
0.3
Polynomial approximation of SPDEs
0.4
0.5
x
0.6
0.7
0.8
0.9
1
RICAM, Linz, 12-16/12/2011
18
Stochastic multivariate polynomial approximation
Collocation on sparse grids
A simple numerical test
Stochastic Collocation approximation:
Gauss-Hermite nodes
isotropic TD grid (m(i) = i; Λ ≡ {i : |i|1 ≤ w })
measured error kE[u] − E[uΛ ]kL2 ρ
comparison with Monte Carlo
Exponential cov.
Fabio Nobile (EPFL & PoliMi)
Gaussian cov.
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
19
Stochastic multivariate polynomial approximation
Optimized algorithms
Optimization of polynomial spaces
Collocation
Galerkin
uΛSG =
X
up (x)ψp (y)
p∈Λ
find uΛSG by Galerkin projection
of the equation on
PΛ = span{ψp , p ∈ Λ}.
uΛSC =
X O
n)
[u].
∆m(i
n
i∈Λ n=1,...,N
Compute uΛSC by collocation on
the corresponding sparse grid
Question: What is the best index set Λ in both cases?
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
20
Stochastic multivariate polynomial approximation
Optimized algorithms
Optimization of polynomial spaces
Collocation
Galerkin
uΛSG =
X
up (x)ψp (y)
p∈Λ
find uΛSG by Galerkin projection
of the equation on
PΛ = span{ψp , p ∈ Λ}.
uΛSC =
X O
n)
[u].
∆m(i
n
i∈Λ n=1,...,N
Compute uΛSC by collocation on
the corresponding sparse grid
Question: What is the best index set Λ in both cases?
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
20
Stochastic multivariate polynomial approximation
Optimized algorithms
Galerkin projection – best M term approximation
Galerkin optimality:
ku − uΛSG kV ⊗L2ρ (Γ) ≤ C
inf
vΛ ∈V ⊗PΛ
ku − vΛ kV ⊗L2ρ (Γ)
Let {ψp , p ∈ NN } be the orthonormal basis of multivariate
Q
polynomials w.r.t. the denisty ρ(y) = Nn=1 ρn (yn ) and vΛ the
truncated expansion of u
X
vΛ =
up ψp ,
up = E[uψp ]
p∈Λ
Parseval’s identity: ku − vΛ k2V ⊗L2ρ (Γ) =
P
p∈Λ
/
kup k2V
Best M terms approximation
The optimal index set Λ of cardinality M is the one that contains the
M largest Legendre coefficients kup kV
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
21
Stochastic multivariate polynomial approximation
Optimized algorithms
Galerkin projection – best M term approximation
Galerkin optimality:
ku − uΛSG kV ⊗L2ρ (Γ) ≤ C
inf
vΛ ∈V ⊗PΛ
ku − vΛ kV ⊗L2ρ (Γ)
Let {ψp , p ∈ NN } be the orthonormal basis of multivariate
Q
polynomials w.r.t. the denisty ρ(y) = Nn=1 ρn (yn ) and vΛ the
truncated expansion of u
X
vΛ =
up ψp ,
up = E[uψp ]
p∈Λ
Parseval’s identity: ku − vΛ k2V ⊗L2ρ (Γ) =
P
p∈Λ
/
kup k2V
Best M terms approximation
The optimal index set Λ of cardinality M is the one that contains the
M largest Legendre coefficients kup kV
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
21
Stochastic multivariate polynomial approximation
Optimized algorithms
Galerkin projection – best M term approximation
Galerkin optimality:
ku − uΛSG kV ⊗L2ρ (Γ) ≤ C
inf
vΛ ∈V ⊗PΛ
ku − vΛ kV ⊗L2ρ (Γ)
Let {ψp , p ∈ NN } be the orthonormal basis of multivariate
Q
polynomials w.r.t. the denisty ρ(y) = Nn=1 ρn (yn ) and vΛ the
truncated expansion of u
X
vΛ =
up ψp ,
up = E[uψp ]
p∈Λ
Parseval’s identity: ku − vΛ k2V ⊗L2ρ (Γ) =
P
p∈Λ
/
kup k2V
Best M terms approximation
The optimal index set Λ of cardinality M is the one that contains the
M largest Legendre coefficients kup kV
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
21
Stochastic multivariate polynomial approximation
Optimized algorithms
Galerkin projection – best M term approximation
Galerkin optimality:
ku − uΛSG kV ⊗L2ρ (Γ) ≤ C
inf
vΛ ∈V ⊗PΛ
ku − vΛ kV ⊗L2ρ (Γ)
Let {ψp , p ∈ NN } be the orthonormal basis of multivariate
Q
polynomials w.r.t. the denisty ρ(y) = Nn=1 ρn (yn ) and vΛ the
truncated expansion of u
X
vΛ =
up ψp ,
up = E[uψp ]
p∈Λ
Parseval’s identity: ku − vΛ k2V ⊗L2ρ (Γ) =
P
p∈Λ
/
kup k2V
Best M terms approximation
The optimal index set Λ of cardinality M is the one that contains the
M largest Legendre coefficients kup kV
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
21
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimate of Fourier coefficients
For the diffusion problem with uniform random variables, the solution
u(y) is analytic in Γ = [−1, 1]N and the following estimate of the
Legendre coefficients holds [Cohen-DeVore-Schwab ’10, Back-N.-Tamellini-Tempone ’11]
kup kV ≤ C0 e −
P
n
gn pn |p|!
(1)
p!
P
Q
for some gn > 0, with |p| = n pn , p! = n pn !.
Then the optimal index set of level w is (TD-FC)
n
o
X
|p|!
≤w
Λ(w ) = p ∈ NN :
gn pn − log
p!
n
In practice, we estimate numerically the rates gn by inexpensive 1D
analyses.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
24
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimate of Fourier coefficients
For the diffusion problem with uniform random variables, the solution
u(y) is analytic in Γ = [−1, 1]N and the following estimate of the
Legendre coefficients holds [Cohen-DeVore-Schwab ’10, Back-N.-Tamellini-Tempone ’11]
kup kV ≤ C0 e −
P
n
gn pn |p|!
(1)
p!
P
Q
for some gn > 0, with |p| = n pn , p! = n pn !.
Then the optimal index set of level w is (TD-FC)
n
o
X
|p|!
Λ(w ) = p ∈ NN :
gn pn − log
≤w
p!
n
In practice, we estimate numerically the rates gn by inexpensive 1D
analyses.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
24
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimate of Fourier coefficients
For the diffusion problem with uniform random variables, the solution
u(y) is analytic in Γ = [−1, 1]N and the following estimate of the
Legendre coefficients holds [Cohen-DeVore-Schwab ’10, Back-N.-Tamellini-Tempone ’11]
kup kV ≤ C0 e −
P
n
gn pn |p|!
(1)
p!
P
Q
for some gn > 0, with |p| = n pn , p! = n pn !.
Then the optimal index set of level w is (TD-FC)
n
o
X
|p|!
Λ(w ) = p ∈ NN :
gn pn − log
≤w
p!
n
In practice, we estimate numerically the rates gn by inexpensive 1D
analyses.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
24
Stochastic multivariate polynomial approximation
Optimized algorithms
Numerical tests
We consider the 1D problem
(
−(a(x, y)u(x, y)0 )0 = 1 x ∈ D = (0, 1), y ∈ Γ
u(0, y) = u(1, y) = 0, y ∈ Γ
with several choices of a(x, y) and compute Θ(u) = u( 12 ).
We compare:
(Aniso) TD space:
(
Λ(w ) =
)
p ∈ NN :
X
gn pn ≤ w
.
n
(Aniso) TD-FC space:
(
Λ(w ) =
Fabio Nobile (EPFL & PoliMi)
N
X
|p|!
p ∈ NN :
gn pn − log
≤w
p!
n=1
Polynomial approximation of SPDEs
)
.
RICAM, Linz, 12-16/12/2011
26
Stochastic multivariate polynomial approximation
Optimized algorithms
Numerical tests
We consider the 1D problem
(
−(a(x, y)u(x, y)0 )0 = 1 x ∈ D = (0, 1), y ∈ Γ
u(0, y) = u(1, y) = 0, y ∈ Γ
with several choices of a(x, y) and compute Θ(u) = u( 12 ).
We compare:
(Aniso) TD space:
(
Λ(w ) =
)
p ∈ NN :
X
gn pn ≤ w
.
n
(Aniso) TD-FC space:
(
Λ(w ) =
Fabio Nobile (EPFL & PoliMi)
N
X
|p|!
p ∈ NN :
gn pn − log
≤w
p!
n=1
Polynomial approximation of SPDEs
)
.
RICAM, Linz, 12-16/12/2011
26
Stochastic multivariate polynomial approximation
Optimized algorithms
A numerical check: a(x, y) = 1 + 0.1xy1 + 0.5x 2y2
−1
10
Legendre coeff
TD appr
TD−FC appr
−2
10
−4
10
−6
10
best M term
TD
iso−TD
TD−FC
−2
10
−3
10
−4
−8
10
−10
10
10
−5
10
−12
−6
10
10
−14
10
−7
10
−16
10
−8
10
−18
10
−9
−20
10
0
20
40
60
80
100
Figure: Legendre coeffs of Θ(u)
in lexicographic order, with TD
and TD-FC estimates
10
0
20
40
60
80
Figure: Convergence plot for
kΘ(u) − Θ(uM )k2L2ρ (Γ) w.r.t.
M = |Λ|
The Legendre coefficients have been computed with a sufficiently high
level sparse grids.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
27
Stochastic multivariate polynomial approximation
Optimized algorithms
Optimization of sparse grids
uM =
SΛm [u]
=
N
XO
n)
∆m(i
[u].
n
i∈Λ n=1
We use a knapsack problem-approach [Griebel-Knapek ’09, Gerstner-Griebel ’03,
Bungartz-Griebel ’04]: for each multiindex i estimate
∆E (i): how much error decreases if i is added to Λ (error
contribution)
∆W (i): how much work, i.e. number of evaluations, increases if
i is added to Λ (work contribution)
Then estimate the profit of each i as
∆E (i)
P(i) =
∆W (i)
and build the sparse grid using the set Λ of the M indices with the
largest profit.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
28
Stochastic multivariate polynomial approximation
Optimized algorithms
Optimization of sparse grids
uM =
SΛm [u]
=
N
XO
n)
∆m(i
[u].
n
i∈Λ n=1
We use a knapsack problem-approach [Griebel-Knapek ’09, Gerstner-Griebel ’03,
Bungartz-Griebel ’04]: for each multiindex i estimate
∆E (i): how much error decreases if i is added to Λ (error
contribution)
∆W (i): how much work, i.e. number of evaluations, increases if
i is added to Λ (work contribution)
Then estimate the profit of each i as
∆E (i)
P(i) =
∆W (i)
and build the sparse grid using the set Λ of the M indices with the
largest profit.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
28
Stochastic multivariate polynomial approximation
Optimized algorithms
Optimization of sparse grids
uM =
SΛm [u]
=
N
XO
n)
∆m(i
[u].
n
i∈Λ n=1
We use a knapsack problem-approach [Griebel-Knapek ’09, Gerstner-Griebel ’03,
Bungartz-Griebel ’04]: for each multiindex i estimate
∆E (i): how much error decreases if i is added to Λ (error
contribution)
∆W (i): how much work, i.e. number of evaluations, increases if
i is added to Λ (work contribution)
Then estimate the profit of each i as
∆E (i)
P(i) =
∆W (i)
and build the sparse grid using the set Λ of the M indices with the
largest profit.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
28
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimates for ∆W and ∆E
1
∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. in
N
O
n)
∆m(i
n
=
n=1
2
N
Y
( m(in ) − m(in − 1) )
n=1
∆E (i): we use the heuristic
argument: use expansion on
P
orthnormal basis u = p up ψp
X
∆E (i) = k∆m(i) [u]kV ⊗L2ρ = k
up ∆m(i) ψp kV ⊗L2ρ
p
≤
X
kup kV k∆m(i) ψp kL2ρ ≈ kum(i−1) kV k∆m(i) ψm(i−1) kL2ρ
p≥m(i−1)
where kum(i−1) kV : estimated with a-priori / a-posteriori
L(m(i)) := k∆m(i) ψm(i−1) kL2ρ : estimated numerically
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
29
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimates for ∆W and ∆E
1
∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. in
N
O
n)
∆m(i
n
=
n=1
2
N
Y
( m(in ) − m(in − 1) )
n=1
∆E (i): we use the heuristic
argument: use expansion on
P
orthnormal basis u = p up ψp
X
∆E (i) = k∆m(i) [u]kV ⊗L2ρ = k
up ∆m(i) ψp kV ⊗L2ρ
p
≤
X
kup kV k∆m(i) ψp kL2ρ ≈ kum(i−1) kV k∆m(i) ψm(i−1) kL2ρ
p≥m(i−1)
where kum(i−1) kV : estimated with a-priori / a-posteriori
L(m(i)) := k∆m(i) ψm(i−1) kL2ρ : estimated numerically
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
29
Stochastic multivariate polynomial approximation
Optimized algorithms
Estimates for ∆W and ∆E
1
∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. in
N
O
n)
∆m(i
n
=
n=1
2
N
Y
( m(in ) − m(in − 1) )
n=1
∆E (i): we use the heuristic
argument: use expansion on
P
orthnormal basis u = p up ψp
X
∆E (i) = k∆m(i) [u]kV ⊗L2ρ = k
up ∆m(i) ψp kV ⊗L2ρ
p
≤
X
kup kV k∆m(i) ψp kL2ρ ≈ kum(i−1) kV k∆m(i) ψm(i−1) kL2ρ
p≥m(i−1)
where kum(i−1) kV : estimated with a-priori / a-posteriori
L(m(i)) := k∆m(i) ψm(i−1) kL2ρ : estimated numerically
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
29
Stochastic multivariate polynomial approximation
Optimized algorithms
All the pieces together
Optimal index set
(
Λ(w ) =
i ∈ NN
+:
N
X
m(in − 1)gn − log
i=n
N
X
n=1
|m(i − 1)|!
−
m(i − 1)!
L(m(in ))
log
≤w
m(in ) − m(in − 1)
)
(EW - Error Work grids)
where
Legendre coeff + L(m(i)) = error estimate
work estimate
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
30
Stochastic multivariate polynomial approximation
Optimized algorithms
A numerical check: a(x, y) = 1 + 0.1xy1 + 0.5x 2y2
−1
10
0
∆E(i)
Legm(i−1)[ψ(u)]
10
−3
Legm(i−1)[ψ(u)] ⋅ Leb[m(i)]
10
−6
iso SM
EW
adaptive
best M terms
−3
10
−5
10
10
−9
10
−7
10
−12
10
−9
10
−15
10
0
5
10
15
20
25
30
Figure: Estimates of ∆E (i) for
Θ(u)] in lexicographic order
0
20
40
60
80
100
120
140
Figure: Convergence plot for
kΘ(u) − Θ(uM )k2L2ρ (Γ) w.r.t. |Λ|
Nested Clenshaw-Curtis knots
The terms ∆E (i) have been computed with a sufficiently high
level sparse grids.
Comparison with dimension adaptive algorithm
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
[Gerstner-Griebel ’03, Klimke, PhD ’06]
RICAM, Linz, 12-16/12/2011
31
Stochastic multivariate polynomial approximation
Numerical results
Numerical test - 1D stationary lognormal field
L = 1, D = [0, L]2 .


−∇ · a(y, x)∇u(y, x) = 0
u = 1 on x = 0, h = 0 on x = 1


no flux otherwise
a(x, y) = e γ(x,y)
µγ (x) = 0
Cov γ (x, x0 ) = σ 2 e −
|x1 −x01 |2
LC 2
We approximate γ as
γ(y, x) ≈ µ(x) + σa0 y0 + σ
K
X
k=1
h
π
π
i
ak y2k−1 cos
kx1 + y2k sin
kx1
L
L
with yi ∼ N (0, 1), i.i.d.
|z|2
Given the Fourier series σ 2 e − LC 2 =
Fabio Nobile (EPFL & PoliMi)
P∞
k=0 ck
cos
π
L kz
Polynomial approximation of SPDEs
, ak =
√
ck .
RICAM, Linz, 12-16/12/2011
32
Stochastic multivariate polynomial approximation
Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
"Z
#
L
∂u(·, x)
E[Φ(u)],
with Φ =
k(·, x)
dx
∂x
0
Convergence: |E[Φ(uSG )] − E[Φ(u)]|
We compare Monte Carlo estimate with Knapsack grids based on
Gauss-Hermite-Patterson points (nested Gauss-Hermite)
Estimate of Hermite coefficients decay:
for the simpler problem ∇ · a(y)∇u(y, x) = f ,
PN
in
a(y) = e b0 + n=1 yn bn , we have kui kV = C √bni ! .
Qn
Heuristic: use the same ansaz kui kV ≈ C N
n=1
estimate the rates gn numerically.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
−gn in
e√
in !
but
RICAM, Linz, 12-16/12/2011
33
Stochastic multivariate polynomial approximation
Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
"Z
#
L
∂u(·, x)
E[Φ(u)],
with Φ =
k(·, x)
dx
∂x
0
Convergence: |E[Φ(uSG )] − E[Φ(u)]|
We compare Monte Carlo estimate with Knapsack grids based on
Gauss-Hermite-Patterson points (nested Gauss-Hermite)
Estimate of Hermite coefficients decay:
for the simpler problem ∇ · a(y)∇u(y, x) = f ,
PN
in
a(y) = e b0 + n=1 yn bn , we have kui kV = C √bni ! .
Qn
Heuristic: use the same ansaz kui kV ≈ C N
n=1
estimate the rates gn numerically.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
−gn in
e√
in !
but
RICAM, Linz, 12-16/12/2011
33
Stochastic multivariate polynomial approximation
Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
"Z
#
L
∂u(·, x)
E[Φ(u)],
with Φ =
k(·, x)
dx
∂x
0
Convergence: |E[Φ(uSG )] − E[Φ(u)]|
We compare Monte Carlo estimate with Knapsack grids based on
Gauss-Hermite-Patterson points (nested Gauss-Hermite)
Estimate of Hermite coefficients decay:
for the simpler problem ∇ · a(y)∇u(y, x) = f ,
PN
in
a(y) = e b0 + n=1 yn bn , we have kui kV = C √bni ! .
Qn
Heuristic: use the same ansaz kui kV ≈ C N
n=1
estimate the rates gn numerically.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
−gn in
e√
in !
but
RICAM, Linz, 12-16/12/2011
33
Stochastic multivariate polynomial approximation
Numerical results
Numerical test - 1D stationary lognormal field
Correlation length: LC = 0.2
Standard deviation: σ = 0.3 (c.o.v. ∼ 30%)
0
10
−2
10
4
11
−4
14
10
17
sparse grid
−6
1/N0.5
10
1/N1.7
19
MC run1
10
24
21
MC run2
−8
MC run3
29
MC run4
−10
10
0
10
1
10
2
10
3
10
4
10
5
10
The optimal set construction automatically adds new variables
when needed.
No need to truncate a-priori the random field
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
34
Stochastic multivariate polynomial approximation
Numerical results
How the sparse grid looks like
η1
5
5
5
5
5
0
0
0
0
0
−5
−5
0
η
2
−5
5 −5
5
0
−5
−5
0
0
η3
−5
5 −5
5
0
−5
5 −5
5
0
−5
5 −5
5
0
0
−5
5 −5
5
0
−5
−5
0
η4
0
−5
5 −5
5
0
0
−5
5 −5
5
0
−5
5 −5
5
0
0
−5
−5
5
0
0
0
−5
5 −5
5
0
0
−5
5 −5
5
0
−5
5 −5
5
0
0
η5
−5
5 −5
5
0
−5
5 −5
5
0
−5
−5
0
5
0
5
0
5
0
5
0
5
0
η7
5
0
0
0
−5
5 −5
5
0
−5
5 −5
5
0
η6
−5
−5
η2
Fabio Nobile (EPFL & PoliMi)
η3
η4
η5
Polynomial approximation of SPDEs
η6
RICAM, Linz, 12-16/12/2011
35
Polynomial approximation by discrete projection on random points
Outline
1
Elliptic PDE with random coefficients
2
Stochastic multivariate polynomial approximation
Galerkin projection
Collocation on sparse grids
Optimized algorithms
Numerical results
3
Polynomial approximation by discrete projection on random points
4
Conclusions
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
38
Polynomial approximation by discrete projection on random points
Discrete L2 projection using random evaluations
(see poster 11 – G. Migliorati)
Another way to compute a polynomial approximation (besides
Galerkin and sparse grid collocation) consists in doing a discrete least
square approx. using random evaluations (see e.g. [Burkardt-Eldred
2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008, ...])
1
2
3
Generate M random i.i.d. samples yk ∈ Γ, k = 1, . . . , M
Compute the corresponding solutions uk = u(y(k) )
Find the discrete least square approximation ΠΛ,ω
M u ∈ V ⊗ PΛ (Γ)
M
1 X
kuk − v (y(k) )k2V
M
v ∈V ⊗PΛ (Γ)
k=1
ΠΛ,ω
M u = argmin
Two relevant questions
For a given set Λ, how many samples should one use?
What is the accuracy of the random discrete least square
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
39
Polynomial approximation by discrete projection on random points
Discrete L2 projection using random evaluations
(see poster 11 – G. Migliorati)
Another way to compute a polynomial approximation (besides
Galerkin and sparse grid collocation) consists in doing a discrete least
square approx. using random evaluations (see e.g. [Burkardt-Eldred
2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008, ...])
1
2
3
Generate M random i.i.d. samples yk ∈ Γ, k = 1, . . . , M
Compute the corresponding solutions uk = u(y(k) )
Find the discrete least square approximation ΠΛ,ω
M u ∈ V ⊗ PΛ (Γ)
M
1 X
kuk − v (y(k) )k2V
M
v ∈V ⊗PΛ (Γ)
k=1
ΠΛ,ω
M u = argmin
Two relevant questions
For a given set Λ, how many samples should one use?
What is the accuracy of the random discrete least square
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
39
Polynomial approximation by discrete projection on random points
Discrete L2 projection using random evaluations
(see poster 11 – G. Migliorati)
Another way to compute a polynomial approximation (besides
Galerkin and sparse grid collocation) consists in doing a discrete least
square approx. using random evaluations (see e.g. [Burkardt-Eldred
2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008, ...])
1
2
3
Generate M random i.i.d. samples yk ∈ Γ, k = 1, . . . , M
Compute the corresponding solutions uk = u(y(k) )
Find the discrete least square approximation ΠΛ,ω
M u ∈ V ⊗ PΛ (Γ)
M
1 X
kuk − v (y(k) )k2V
M
v ∈V ⊗PΛ (Γ)
k=1
ΠΛ,ω
M u = argmin
Two relevant questions
For a given set Λ, how many samples should one use?
What is the accuracy of the random discrete least square
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
39
Polynomial approximation by discrete projection on random points
Some theoretical results
[Migliorati-N.-von Schwerin-Tempone ’11]
For functions φ : Γ ⊂ RN → R, define
R
continuous norm: kφk2L2ρ = Γ v 2 (y)ρ(y)dy
P
2
discrete norm: kφk2M,ω = M1 M
i=1 φ(yi ) , with yi ∼ ρ(y)dy, i.i.d.
random discrete least square projection: ΠΛ,ω
M φ ∈ PΛ (Γ),
2
ΠΛ,ω
M φ = argmin kφ − v kM,ω .
v ∈PΛ (Γ)
Theorem 1
ω
Let C (M, Λ) := sup
v ∈PΛ (Γ)
1
2
kv k2L2ρ
kv k2M,ω
. Then
C ω (M, Λ) → 1 almost surely when M → ∞
p
2 ≤ (1 +
kφ − ΠΛ,ω
φk
C ω (M, Λ)) inf v ∈PΛ (Γ) kφ − v kL∞
L
M
ρ
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
40
Polynomial approximation by discrete projection on random points
Some theoretical results
[Migliorati-N.-von Schwerin-Tempone ’11]
So far we have only a result for 1D functions, Γ = [−1, 1], uniform
distribution and approx. in the polynomial space Pw of degree w.
Theorem 2
For any α ∈ (0, 1), let M be such that
Then, it holds
r
w,ω
P kφ − ΠM
φkL2ρ ≤
1+2
M
3 log((M+1)/α)
M +1
3 log
α
√
= 4 3 w2
!
!
inf kφ − v kL∞
v ∈Pw
≥ 1 − α.
Notice that log((M + 1)/α) ≈ log (C w2 /α)
These results are confirmed numerically; the high dimentional
case seems more foregiving w.r.t. the constraint M ∼ #Λ2
To achieve an approximation in PΛ , does this technique need less
sampling points than the corresponding sparse grid ?
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
41
Polynomial approximation by discrete projection on random points
Some theoretical results
[Migliorati-N.-von Schwerin-Tempone ’11]
So far we have only a result for 1D functions, Γ = [−1, 1], uniform
distribution and approx. in the polynomial space Pw of degree w.
Theorem 2
For any α ∈ (0, 1), let M be such that
Then, it holds
r
w,ω
P kφ − ΠM
φkL2ρ ≤
1+2
M
3 log((M+1)/α)
M +1
3 log
α
√
= 4 3 w2
!
!
inf kφ − v kL∞
v ∈Pw
≥ 1 − α.
Notice that log((M + 1)/α) ≈ log (C w2 /α)
These results are confirmed numerically; the high dimentional
case seems more foregiving w.r.t. the constraint M ∼ #Λ2
To achieve an approximation in PΛ , does this technique need less
sampling points than the corresponding sparse grid ?
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
41
Polynomial approximation by discrete projection on random points
Some theoretical results
[Migliorati-N.-von Schwerin-Tempone ’11]
So far we have only a result for 1D functions, Γ = [−1, 1], uniform
distribution and approx. in the polynomial space Pw of degree w.
Theorem 2
For any α ∈ (0, 1), let M be such that
Then, it holds
r
w,ω
P kφ − ΠM
φkL2ρ ≤
1+2
M
3 log((M+1)/α)
M +1
3 log
α
√
= 4 3 w2
!
!
inf kφ − v kL∞
v ∈Pw
≥ 1 − α.
Notice that log((M + 1)/α) ≈ log (C w2 /α)
These results are confirmed numerically; the high dimentional
case seems more foregiving w.r.t. the constraint M ∼ #Λ2
To achieve an approximation in PΛ , does this technique need less
sampling points than the corresponding sparse grid ?
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
41
Conclusions
Outline
1
Elliptic PDE with random coefficients
2
Stochastic multivariate polynomial approximation
Galerkin projection
Collocation on sparse grids
Optimized algorithms
Numerical results
3
Polynomial approximation by discrete projection on random points
4
Conclusions
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
48
Conclusions
Conclusions
Solutions to elliptic equations with random coefficients typically
feature analytic dependence on the parameters. Polynomial
approximations are very effective.
Sharp a-priori / a-posteriori analysis of the decay of the expansion of
the solution in polynomial chaos allows to construct optimized
polynomial spaces / sparse grids that provide effective
approximations also in the infinite dimensional case.
Discrete least square projection using random evaluations is a
possible alternative to Galerkin or Collocation approaches. However,
a better understanding is needed on the stability of the projection
and the correct number of samples to use.
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
49
Conclusions
References
G. Migliorati and F. Nobile and E. von Schwrin and R. Tempone
Analysis of the discrete L2 projection on polynomial spaces with random evaluations,
submitted. Available as MOX-Report.
J. Bäck and F. Nobile and L. Tamellini and R. Tempone
On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation
methods, MOX Report 23/2011, to appear in M3AS.
I. Babuška, F. Nobile and R. Tempone.
A stochastic collocation method for elliptic PDEs with random input data, SIAM Review,
52(2):317–355, 2010
F. Nobile and R. Tempone
Analysis and implementation issues for the numerical approximation of parabolic equations
with random coefficients, IJNME, 80:979–1006, 2009
F. Nobile, R. Tempone and C. Webster
An anisotropic sparse grid stochastic collocation method for PDEs with random input
data, SIAM J. Numer. Anal., 46(5):2411–2442, 2008
Fabio Nobile (EPFL & PoliMi)
Polynomial approximation of SPDEs
RICAM, Linz, 12-16/12/2011
50